Answer
Solution set = $\{0\}.$
Work Step by Step
Recognize that $3^{2x}=(3^{x})^{2}$. Introduce a new variable t:
$ t=3^{x}.$
$ t \gt 0 $, because $3^{x}$ is always positive.
We solve
$ t^{2}+t-2=0$
Factor by finding factors of $-2$ whose sum is $+1$:
$(t+2)(t-1)=0$
$ t+2=0$ or $ t-1=0$
$ t=-2$ or $ t=1$
$ t=-2$ is discarded, as $ t \gt 0 $, leaving
$ t=1\qquad $... bring back x
$ 3^{x}=1\qquad $... apply $ a^{0}=1$
$ 3^{x}=3^{0}\qquad $... if $ a^{m}=a^{n}$, then $ m=n $
$ x=0$
Solution set = $\{0\}.$
